Abstract
In this article, we are concerned with a nondifferentiable minimax fractional programming problem. We derive the sufficient condition for an optimal solution to the problem and then establish weak, strong, and strict converse duality theorems for the problem and its dual problem under B(p, r)invexity assumptions. Examples are given to show that B(p, r)invex functions are generalization of (p, r)invex and convex functions
AMS Subject Classification: 90C32; 90C46; 49J35.
Keywords:
nondifferentiable fractional programming; optimality conditions; B(p, r)invex function; duality theorems1 Introduction
The mathematical programming problem in which the objective function is a ratio of two numerical functions is called a fractional programming problem. Fractional programming is used in various fields of study. Most extensively, it is used in business and economic situations, mainly in the situations of deficit of financial resources. Fractional programming problems have arisen in multiobjective programming [1,2], game theory [3], and goal programming [4]. Problems of these type have been the subject of immense interest in the past few years.
The necessary and sufficient conditions for generalized minimax programming were first developed by Schmitendorf [5]. Tanimoto [6] applied these optimality conditions to define a dual problem and derived duality theorems. Bector and Bhatia [7] relaxed the convexity assumptions in the sufficient optimality condition in [5] and also employed the optimality conditions to construct several dual models which involve pseudoconvex and quasiconvex functions, and derived weak and strong duality theorems. Yadav and Mukhrjee [8] established the optimality conditions to construct the two dual problems and derived duality theorems for differentiable fractional minimax programming. Chandra and Kumar [9] pointed out that the formulation of Yadav and Mukhrjee [8] has some omissions and inconsistencies and they constructed two modified dual problems and proved duality theorems for differentiable fractional minimax programming.
Lai et al. [10] established necessary and sufficient optimality conditions for nondifferentiable minimax fractional problem with generalized convexity and applied these optimality conditions to construct a parametric dual model and also discussed duality theorems. Lai and Lee [11] obtained duality theorems for two parameterfree dual models of nondifferentiable minimax fractional problem involving generalized convexity assumptions.
Convexity plays an important role in deriving sufficient conditions and duality for nonlinear programming problems. Hanson [12] introduced the concept of invexity and established KarushKuhnTucker type sufficient optimality conditions for nonlinear programming problems. These functions were named invex by Craven [13]. Generalized invexity and duality for multiobjective programming problems are discussed in [14], and inseparable Hilbert spaces are studied by Soleimanidamaneh [15]. Soleimanidamaneh [16] provides a family of linear infinite problems or linear semiinfinite problems to characterize the optimality of nonlinear optimization problems. Recently, Antczak [17] proved optimality conditions for a class of generalized fractional minimax programming problems involving B(p, r)invexity functions and established duality theorems for various duality models.
In this article, we are motivated by Lai et al. [10], Lai and Lee [11], and Antczak [17] to discuss sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem with B(p, r)invexity. This article is organized as follows: In Section 2, we give some preliminaries. An example which is B(1, 1)invex but not (p, r)invex is exemplified. We also illustrate another example which (1, 1)invex but convex. In Section 3, we establish the sufficient optimality conditions. Duality results are presented in Section 4.
2 Notations and prelominaries
Definition 1. Let f : X → R (where X ⊆ R^{n}) be differentiable function, and let p, r be arbitrary real numbers. Then f is said to be (p, r)invex (strictly (p, r)invex) with respect to η at u ∈ X on X if there exists a function η : X × X → R^{n }such that, for all x ∈ X, the inequalities
hold.
Definition 2 [17]. The differentiable function f : X → R (where X ⊆ R^{n}) is said to be (strictly) B(p, r)invex with respect to η and b at u ∈ X on X if there exists a function η : X × X → R^{n }and a function b : X × X → R_{+ }such that, for all x ∈ X, the following inequalities
hold. f is said to be (strictly) B(p, r)invex with respect to η and b on X if it is B(p, r)invex with respect to same η and b at each u ∈ X on X.
Remark 1 [17]. It should be pointed out that the exponentials appearing on the righthand sides of the inequalities above are understood to be taken componentwise and 1 = (1, 1, ..., 1) ∈ R^{n}.
Example 1. Let X = [8.75, 9.15] ⊂ R. Consider the function f : X → R defined by
Let η : X × X → R be given by
To prove that f is (1, 1)invex, we have to show that
Now, consider
as can be seen form Figure 1.
Figure 1. φ = sin^{2}x + sin 2u(e^{12(1+u) } 1)  sin^{2}u.
Hence, f is (1, 1)invex.
Further, for x = 8.8 and u = 9.1, we have
Thus f is not convex function on X.
Example 2. Let X = [0.25, 0.45] ⊂ R. Consider the function f : X → R defined by
Let η : X × X → R and b : X × X → R_{+ }be given by
and
respectively.
The function f defined above is B(1, 1)invex as
as can be seen from Figure 2.
Figure 2.
However, it is not (p, r) invex for all p, r ∈ (10^{17}, 10^{17}) as
(for x = 0.4 and u = 0.42)
< 0 as can be seen from Figure 3.
Figure 3.
Hence f is B(1, 1)invex but not (p, r)invex.
In this article, we consider the following nondifferentiable minimax fractional programming problem:
(FP)
where Y is a compact subset of R^{m}, l(., .): R^{n }× R^{m }→ R, m(., .): R^{n }× R^{m }→ R, are C^{1 }functions on R^{n }× R^{m }and g(.): R^{n }→ R^{p }is C^{1 }function on R^{n}. D and E are n × n positive semidefinite matrices.
Let S = {x ∈ X : g(x) ≤ 0} denote the set of all feasible solutions of (FP).
Any point x ∈ S is called the feasible point of (FP). For each (x, y) ∈ R^{n }× R^{m}, we define
such that for each (x, y) ∈ S × Y,
For each x ∈ S, we define
where
with
Since l and m are continuously differentiable and Y is compact in R^{m}, it follows that for each x* ∈ S, Y (x*) ≠ ∅, and for any
2.1 Generalized Schwartz inequality
Let A be a positivesemidefinite matrix of order n. Then, for all, x, w ∈ R^{n},
Equality holds if for some λ ≥ 0,
Evidently, if
If the functions l, g, and m in problem (FP) are continuously differentiable with respect to x ∈ R^{n}, then Lai et al. [10] derived the following necessary conditions for optimality of (FP).
Theorem 1 (Necessary conditions). If x* is a solution of (FP) satisfying x*^{T}Dx* >0, x*^{T}Ex* > 0, and ∇g_{h}(x*), h ∈ H(x*) are linearly independent, then there exist
Remark 2. All the theorems in this article will be proved only in the case when p ≠ 0, r ≠ 0. The proofs in the other cases are easier than in this one. It follows from the form of inequalities which are given in Definition 2. Moreover, without limiting the generality considerations, we shall assume that r > 0.
3 Sufficient conditions
Under smooth conditions, say, convexity and generalized convexity as well as differentiability, optimality conditions for these problems have been studied in the past few years. The intrinsic presence of nonsmoothness (the necessity to deal with nondifferentiable functions, sets with nonsmooth boundaries, and setvalued mappings) is one of the most characteristic features of modern variational analysis (see [18,19]). Recently, nonsmooth optimizations have been studied by some authors [2023]. The optimality conditions for approximate solutions in multiobjective optimization problems have been studied by Gao et al. [24] and for nondifferentiable multiobjective case by Kim et al. [25]. Now, we prove the sufficient condition for optimality of (FP) under the assumptions of B(p, r)invexity.
Theorem 2 (Sufficient condition). Let x* be a feasible solution of (FP) and there exist a positive integer s, 1 ≤ s ≤ n + 1,
(i)
(ii)
Then x* is an optimal solution of (FP).
Proof. Suppose to the contrary that x* is not an optimal solution of (FP). Then there exists an
We note that
for
Thus, we have
It follows that
From (1), (3), (5), (6) and (7), we obtain
It follows that
As
holds for all x ∈ S, and so for
From the feasibility of
By B_{g}(p, r)invexity of
Since b_{g}(x, x*) ≥ 0 for all x ∈ S then by (4) and (10), we obtain
By adding the inequalities (9) and (11), we have
which contradicts (2). Hence the result. □
4 Duality results
In this section, we consider the following dual to (FP):
where
If, for a triplet
Let S_{FD }denote a set of all feasible solutions for problem (FD). Moreover, let S_{1 }denote
Now we derive the following weak, strong, and strict converse duality theorems.
Theorem 3 (Weak duality). Let x be a feasible solution of (P) and
(i)
(ii)
Then,
Proof. Suppose to the contrary that
Then, we have
It follows from (5) that
with at least one strict inequality, since t = (t_{1}, t_{2}, ..., t_{s}) ≠ 0.
From (1), (13), (16) and (18), we have
Hence
Since
From (19) and b(x, a) > 0 together with the inequality above, we get
Using the feasibility of x together with μ_{h }≥ 0, h ∈ H, we obtain
From hypothesis (ii), we have
As b_{g}(x, a) ≥ 0 then by (14) and (21), we obtain
Thus, by (20) and (22), we obtain the inequality
which contradicts (12). Hence (17) holds. □
Theorem 4 (Strong duality). Let x* be an optimal solution of (FP) and ∇g_{h}(x*), h ∈ H(x*) is linearly independent. Then there exist
Proof. If x* be an optimal solution of (FP) and ∇g_{h}(x*), h ∈ H(x*) is linearly independent, then by Theorem 1, there exist
The optimality of this feasible solution for (FD) thus follows from Theorem 3. □
Theorem 5 (Strict converse duality). Let x* and
Proof. We shall assume that
By feasibility of x* together with μ_{h }≥ 0, h ∈ H, we obtain
By assumption,
Then, from Definition 2, we get
Therefore, by (25), we obtain the inequality
As
From the hypothesis
Therefore, by (13),
Since t_{i }≥ 0, i = 1, 2, ..., s, therefore there exists i* such that
Hence, we obtain the following inequality
which contradicts (23). Hence the results. □
5 Concluding remarks
It is not clear that whether duality in nondifferentiable minimax fractional programming with B(p, r)invexity can be further extended to secondorder case.
6 Competing interests
The authors declare that they have no competing interests.
7 Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
Izhar Ahmad thanks the King Fahd University of Petroleum and Minerals for the support under the Fast Track Project no. FT100023. Ravi P. Agarwal gratefully acknowledges the support provided by the King Fahd University of Petroleum and Minerals to carry out this research. The authors wish to thank the referees for their several valuable suggestions which have considerably improved the presentation of this article.
References

Gulati, TR, Ahmad, I: Efficiency and duality in multiobjective fractional programming. Opsearch. 32, 31–43 (1990)

Weir, T: A dual for multiobjective fractional programming. J Inf Optim Sci. 7, 261–269 (1986)

Chandra, S, Craven, BD, Mond, B: Generalized fractional programming duality: a ratio game approach. J Aust Math Soc B. 28, 170–180 (1986). Publisher Full Text

Charnes, A, Cooper, WW: Goal programming and multiobjective optimization, Part I. Eur J Oper Res. 1, 39–54 (1977). Publisher Full Text

Schmitendorf, WE: Necessary conditions and sufficient optimality conditions for static minimax problems. J Math Anal Appl. 57, 683–693 (1977). Publisher Full Text

Tanimoto, S: Duality for a class of nondifferentiable mathematical programming problems. J Math Anal Appl. 79, 283–294 (1981)

Bector, CR, Bhatia, BL: Sufficient optimality and duality for a minimax problems. Utilitas Mathematica. 27, 229–247 (1985)

Yadav, SR, Mukherjee, RN: Duality for fractional minimax programming problems. J Aust Math Soc B. 31, 484–492 (1990). Publisher Full Text

Chandra, S, Kumar, V: Duality in fractional minimax programming. J Aust Math Soc A. 58, 376–386 (1995). Publisher Full Text

Lai, HC, Liu, JC, Tanaka, K: Necessary and sufficient conditions for minimax fractional programming. J Math Anal Appl. 230, 311–328 (1999). Publisher Full Text

Lai, HC, Lee, JC: On duality theorems for a nondifferentiable minimax fractional programming. J Comput Appl Math. 146, 115–126 (2002). Publisher Full Text

Hanson, MA: On sufficiency of the KuhnTucker conditions. J Math Anal Appl. 80, 545–550 (1981). Publisher Full Text

Craven, BD: Invex functions and constrained local minima. Bull Aust Math Soc. 24, 357–366 (1981). Publisher Full Text

Aghezzaf, B, Hachimi, M: Generalized invexity and duality in multiobjective programming problems. J Global Optim. 18, 91–101 (2000). Publisher Full Text

Soleimanidamaneh, M: Generalized invexity in separable Hilbert spaces. Topology. 48, 66–79 (2009). Publisher Full Text

Soleimanidamaneh, M: Infinite (semiinfinite) problems to characterize the optimality of nonlinear optimization problems. Eur J Oper Res. 188, 49–56 (2008). Publisher Full Text

Antczak, T: Generalized fractional minimax programming with B(p, r)invexity. Comput Math Appl. 56, 1505–1525 (2008). Publisher Full Text

Mordukhovich, BS: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Grundlehren Series (Fundamental Principles of Mathematical Sciences) (2006)

Mordukhovich, BS: Variations Analysis and Generalized Differentiation, II: Applications. Springer, Grundlehren Series (Fundamental Principles of Mathematical Sciences) (2006)

Agarwal, RP, Ahmad, I, Husain, Z, Jayswal, A: Optimality and duality in nonsmooth multiobjective optimization involving Vtype I invex functions. J Inequal Appl. 2010, Article ID 898626 (2010) 14

Kim, DS, Lee, HJ: Optimality conditions and duality in nonsmooth multiobjective programs. J Inequal Appl. 2010, Article ID 939537 (2010) 12

Soleimanidamaneh, M: Nonsmooth optimization using Mordukhovich's subdifferential. SIAM J Control Optim. 48, 3403–3432 (2010). Publisher Full Text

Soleimanidamaneh, M, Nieto, JJ: Nonsmooth multipleobjective optimization in separable Hilbert spaces. Nonlinear Anal. 71, 4553–4558 (2009). Publisher Full Text

Gao, Y, Yang, X, Lee, HWJ: Optimality conditions for approximate solutions in multiobjective optimization problems. J Inequal Appl. 2010, Article ID 620928 (2010) 17

Kim, HJ, Seo, YY, Kim, DS: Optimality conditions in nondifferentiable Ginvex multiobjective programming. J Inequal Appl. 2010, Article ID 172059 (2010) 13